In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of


0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.
6 3 2 3 6 6 3 2 3 6


Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.
sec 11𝜋/6

Verified step by step guidance

1. Identify the angle given: here, the angle is \( t = \frac{11\pi}{6} \). This corresponds to a point on the unit circle.
2. Recall that the secant function is defined as the reciprocal of the cosine function: \( \sec t = \frac{1}{\cos t} \).
3. Locate the coordinates \((x, y)\) of the point on the unit circle corresponding to \( t = \frac{11\pi}{6} \). The \( x \)-coordinate represents \( \cos t \) and the \( y \)-coordinate represents \( \sin t \).
4. Calculate \( \sec \frac{11\pi}{6} \) by taking the reciprocal of the \( x \)-coordinate (cosine value) found in the previous step: \( \sec \frac{11\pi}{6} = \frac{1}{x} \).
5. Check if the cosine value is zero. If it is, then \( \sec t \) is undefined because division by zero is not possible.

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, where one full rotation equals 2π radians. Dividing the circle into equal arcs corresponds to specific radian values, which help locate points (x, y) representing cosine and sine of the angle.

Trigonometric Functions and Coordinates

On the unit circle, the x-coordinate of a point corresponds to the cosine of the angle t, and the y-coordinate corresponds to the sine of t. Other trigonometric functions like secant (sec) are defined in terms of sine and cosine; for example, sec t = 1/cos t. Understanding these relationships allows evaluation of trig functions using coordinates.

Domain and Undefined Values of Trigonometric Functions

Some trigonometric functions are undefined at certain angles where their denominators are zero. For sec t = 1/cos t, the function is undefined when cos t = 0. Identifying these points on the unit circle is essential to determine when a function value does not exist or is undefined.